Method to determine tdc in an internal combustion engine

ABSTRACT

A method and apparatus for accurately determining TTX: in a cylinder of an internal combustion engine. Gas pressure is measured in the cylinder relative to crank angle, giving rise to a curve of gas pressure with crank angle. An offset is found by first computing the angular position of an inflexion point of the compression pressure curve measured as a function of the crank angle, and then computing the offset as the difference between the angular position of a piston in the cylinder and a theoretical value for said angular position of the piston dependent on a known or measured flywheel angle. The value of TDC corrected by the offset is then used to calculate a value of work output such as the indicated Mean Effective Pressure (IMEP) which may be used for diagnostic and/or control purposes.

TECHNICAL AREA

[0001] The present invention concerns the technical area of methods tocalculate variables useful in performing diagnostics and control of acombustion engine, such as the Indicated Mean Effective Pressure (IMEP),and other variables which require that the crank angle of the piston inthe cylinder is accurately known. In particular the invention concerns amethod to determine accurately when a piston in a cylinder of aninternal combustion engine is at top dead centre (TDC).

TECHNICAL BACKGROUND

[0002] Many methods exist for controlling operation of a combustionengine. In particular, measurements of pressure from one or morecylinders of an engine are commonly used as a basis for engineperformance measurement, monitoring and for control of an engine.

[0003] One area of use for continually measuring pressure sensors is invery large combustion engines which are operated at relatively lowrevolutions per minute. Such motors are used for example as shipsengines and as stationary engines for driving electrical generators andgas compressors. However engine monitoring and control is alsoincreasingly applied to smaller engines of the type such as those usedin ordinary vehicles such as cars, trucks and buses.

[0004] Because of the growing demands for reduced consumption of fueland continually increasing environmental demands on the chemicalcomposition of exhaust gases the requirement to monitor the operation ofcombustion engines has increased. Misfiring influences exhaust gaschemical composition and can also negatively influence the working lifeof a combustion engine. With the help of continuous measurementmisfiring and other factors affecting engine performance can be detectedand action be taken to ensure proper functioning is regained.

[0005] A difficulty in the calculation of work output, IMEP of acylinder is to determine accurately a crank angle for a piston togetherwith the pressure corresponding to that crank angle. Generally crankangle of a given piston is calculated from a measurement of the angularposition of the crankshaft of the engine (crankshaft angle), as it isdifficult to measure the piston position directly in the cylinder duringoperation. The crankshaft angle is usually obtained by an independentmeasurement of the angular position of the flywheel (flywheel angle)relative to a known position such as a timing mark, according to knownmethods.

[0006] Top dead centre (TDC) of a piston in a cylinder may be determinedmechanically according to known methods by measuring with, for example,a capacitive sensor or a position transducer in an engine at rest.Another method includes shutting off fuel to a given cylinder of a motorduring operation, measuring the cylinder pressure, and estimating theposition of TDC in that cylinder from an expected symmetry of themeasured pressure curve. In this method a maximum of the symmetricalpressure curve indicates TDC, although that indicated maximum point maycontain thermodynamical errors. A principal disadvantage with both ofthese methods is that they do not provide for determination of TDC in amotor under normal operation.

[0007] A method and apparatus described in JP 9329049 discloses that avalue for IMEP may be calculated dependent on measured cylinder pressureand known crankshaft angle, and that such a value may be used forcontrol or regulation of fuel supply amount and ignition timing for amotor vehicle engine.

[0008] What is difficult in practice with most methods described is todetermine accurately an angular offset that usually exists between theflywheel angle and the piston crank angle at a given moment in arevolution. In order to determine the offset it is usually consideredsufficient to determine the flywheel angle in relation to a knownposition of the piston in the cylinder, and the position commonly usedis the top dead centre position in the cylinder. The TDC positioncorresponds to a crank angle for a piston of 0 degrees.

[0009] However, the usual methods rely on a measurement based on a pointmarked at the periphery of the flywheel, which measurement typicallydepends on a position of a train of mechanically connected enginecomponents in an engine at rest.

[0010] For instance in EP 742 359 A2, a method and apparatus forcontrolling the operation of an internal combustion engine is described.In this description the crankshaft angle is measured by a sensordetecting gear teeth on a ring gear fixed to the flywheel of an engine.

[0011] However manufacturing tolerances of engine parts mean in practicethat dimensions of parts vary somewhat from specified or expecteddimensions. In addition in an engine under operation the dynamicinterplay of stresses on components, free play clearances betweenmechanically connected components, friction and inertia of componentsmeans that the mechanical distances and relationships between componentsunder operation may not be the same as distances and relationshipsbetween the same components at rest. The relative position ofmechanically connected piston and a flywheel at rest does not provide ameasurement of the relative position of the same piston and flywheel, asa basis for a computation of the crank angle during operation of anengine, to a degree which is sufficiently accurate for modern enginediagnostic systems or engine control systems.

SUMMARY OF THE INVENTION

[0012] It is an object of the invention to provide an accuratedetermination of position of a piston in a cylinder of an internalcombustion engine, a position such as TDC, under normal operation of theengine.

[0013] Embodiments of the present invention aim to address one orseveral of the above mentioned problems. According to one aspect of theinvention a method is provided for producing information about theposition of a piston in a cylinder with respect to crankshaft angle bymeasuring gas pressure in the cylinder with respect to crankshaft angleand computing an inflexion point in a curve of gas pressure againstcrankshaft angle. The position given by the computed inflexion point iscompared to the position obtained by a theoretical calculation based ona known or measured flywheel or crankshaft angle, and an offset found.The offset is then used to calculate an accurate value for pistonposition such as TDC and to compute variables such as IMEP.

[0014] In another aspect of the invention a system is provided todetermine a position of a piston in a cylinder, including a gas pressuresensor and a computation means suitably arranged to measure gas pressurein a cylinder with respect to a crankshaft angle of a motor, compute aninflexion point of a curve of gas pressure with respect to crankshaftangle and provide an offset representing the difference between pistonposition determined by the computed inflexion point and piston positioncalculated according to a known or measured crankshaft angle.

[0015] The present invention is directed to determining a point ofinflection in a curve from a measurement of cylinder gas pressure andcrankshaft angle, that is, the point at which a derivative of thepressure curve with respect to crankshaft angle is at a maximum, orequivalently where the second derivative is equal to zero. The sameangular position may also be computed theoretically, by applying wellknown thermodynamic equations together with data for the heat capacitiesof the constituent gases taking part in the compression. By comparingthe measured value with the theoretical value, the difference or offsetbetween the measured angular position and theoretical angular positionof the crankshaft or the flywheel, is obtained and used to calculate thepiston position accurately with respect to TDC (or crankshaft angle).Having accurately determined the local crank angle for each piston thisvalue, together with cylinder pressure, may be further used to calculatea measure of work output such as IMEP for each cylinder of an engine.

[0016] In another embodiment of the invention, IMEP can be calculatedwith reduced accuracy without the need for an independent measurement ofthe flywheel angle. In this embodiment the pressure signal is digitallysampled at known sample times. The derivative of the pressure signal isthen computed with respect to time, which is proportional to thederivative with respect to the flywheel angle, if the rotational speedis constant. The timing of the inflection points thus computed for allcylinders where the pressure is measured, is then used together with theknown geometry of the crank shaft to estimate the flywheel position atthe sample times. This estimated flywheel position is then used tocompute local crank angles for the pistons at the sample times, and usedtogether with the sampled pressure signals to compute variables for useas engine diagnostics, such as IMEP.

[0017] In another aspect of the invention, a computer program product isprovided to carry out the steps of a method according to the invention.

[0018] The advantage of using the angular position of the inflexionpoint of the pressure as a function of the crankshaft angle as thereference point, compared with simply using the angular position of themaximum of the pressure according to the state of the art, is that theinflexion point can be detected during normal operation of the engine.During normal operation the fuel is normally injected at, or slightlyahead of TDC. The maximum of the compression curve is then obliteratedby the ignition of the fuel. Another advantage gained by using theangular position of the inflexion point is that it is virtuallyindependent of the value of the pressure at the start of thecompression, and only weakly dependent on other well known geometricaland thermodynamic quantities. Its value can therefore be computedtheoretically with great accuracy.

[0019] The main advantage of the invention is that TDC is determinedmore accurately with a method and apparatus that is also economic toapply. This means that sophisticated engine diagnostics, control andregulation may be applied to smaller, less expensive engines and notonly engines in larger installations such as generators. These benefitscan be gained both by more effective development of new or improvedengines and from more effective management of engines under operation.The invention consequently offers significant environmental benefits dueto reduced fuel consumption and emissions with reduced environmentalimpact without high investment in engine technology or measurementcosts.

[0020] Another advantage of one embodiment of the invention is that onlyone input is required, an input dependent on gas pressure in a cylinder,in order to calculate IMEP according to the invention. Thus with fewercomponents the system is potentially less prone to breakdowns. Inaddition, a TDC position of the piston in a cylinder of an internalcombustion engine is determined with greater accuracy than existingmethods. It is also advantageous that another embodiment of theinvention provides a means for determining crank angle that isindependent of measurement of the flywheel angle.

[0021] The invention may be applied very widely to any motor, old ornew, that can be fitted with a sensor for gas pressure for at least onecylinder. In an advantageous embodiment, the invention may be used toprovide an input for an engine control or management device or system sothat the engine can be monitored and/or regulated. The calculationelements of the method may be carried out by one or more inexpensiveelectronic components installed in a vehicle such that the results mayadvantageously be used to control fuel economy and exhaust emissionquality.

BRIEF DESCRIPTION OF THE FIGURES

[0022] A more complete understanding of the method and apparatus of thepresent invention may be had by reference to the following detaileddescription when taken in conjunction with the accompanying drawingswherein:

[0023]FIG. 1 shows a schematic curve of a relation between gas pressurein a cylinder and crank angle for a method according to an embodiment ofthe present invention;

[0024]FIG. 2 shows a schematic curve of derivative of a relation betweengas pressure in a cylinder and crank angle according to an embodiment ofthe invention;

[0025]FIG. 3 shows a block diagram of a flow chart for a method todetermine an offset between an expected position of piston relative to ameasured flywheel angle and a position of the piston relative theflywheel angle computed according to an embodiment of the invention;

[0026]FIG. 4 shows a schematic diagram for system to control an engineusing a method according to an embodiment of the invention.

DESCRIPTION OF EMBODIMENTS

[0027] In order to determine a Top Dead Centre position of the piston ina cylinder the present invention comprises a method for determining apoint of inflection in a curve dependent on a measurement of cylinderpressure related to crank angle. That is, the point in the pressurecurve at which a derivative of the pressure curve in relation to crankangle is at a maximum or equivalent, and where a second derivative ofthe pressure curve is equal to zero.

[0028]FIG. 1 shows a schematic graph of a curve 1 for measured cylinderpressure on the y-axis plotted against crank angle in degrees on thex-axis. A first point of inflection 2 may be seen at about −13° on thecurve.

[0029] The position of that point α (in radians) may be determinedtheoretically with the help of a fairly complicated formula as follows:$\quad \begin{matrix}{\alpha = -} \\\sqrt{\frac{4}{\left( {1 + \lambda} \right)\left( {1 + {2\left( {\gamma - 0.0217} \right)}} \right)\left( {r_{c} - 1} \right)} + \frac{\frac{4}{3}\left( {{- 1} - {3\lambda} + {3\lambda^{2}}} \right)\left( {5 + {4\left( {\gamma - 0.0217} \right)}} \right)}{\left( {1 + \lambda} \right)^{2}\left( {1 + {2\left( {\gamma - 0.0217} \right)}} \right)^{3}\left( {r_{c} - 1} \right)^{2}}}\end{matrix}$

[0030] where

[0031] λ=Relation between crank radius and piston connecting rod length.

[0032] γ=Quotient between heat capacity at constant pressure and heatcapacity at constant volume (polytropic coefficient) for the compressedgas at the estimated temperature of the gas at the inflection point.

[0033] r_(c)=Compression ratio of the motor.

[0034] α=theoretical inflexion point in radians.

[0035] The dependency of γ means that it is necessary to know about thecomposition of the compressed gas and its temperature in order for avalue for quotient γ to be calculated. The dependency of γ is, as ithappens, not particularly great. γ is equal to 1.40 for an ideal gas andremains almost always in within an interval of 1.30-1.40 for real gases.An uncertainty in the value of γ of 0.01 units gives an uncertainty ofabout 0.03° in the theoretically determined inflection point.

[0036] The dependency of other parameters is also moderate. An error indetermination of compression ratio of the motor of 1% would give anerror in determination of the inflection point according to the aboveequation of about 0.07°.

[0037] Thermal conductivity between the compressed gas and the cylinderwall changes the measured value of the pressure curve somewhat. Themaximum pressure is reduced and the curve is shifted to the leftcompared to FIG. 1 by a value of the order of 0-0.3°. A correction termfor this effect has also been calculated within the embodiments of theinvention, however it is necessary that the coefficient of thermalconductivity and cylinder wall temperature are both known, which in mostcases is only approximately so. Thus a value for position of TDC givenin radians may be accurately determined under normal operation of amotor by the described method. Once the crank angle has been determined,the IMEP is readily calculated.

[0038]FIG. 2. shows a derivative of the pressure/crank angle graph ofFIG. 1. The maximum 2′ point on the curve 1′ is the derivative of thecorresponding inflection point 2 on curve 1 shown in FIG. 1. Bycalculating the derivative, the inflection point 2 may be clearlyidentified in the curve of FIG. 2 as a maximum.

[0039]FIG. 3 an exemplary flowchart for a method according to anembodiment of the invention is shown. Gas pressure in the cylinder ismeasured, preferably continually, at 4, using any kind of sensor ormeasurement means. The flywheel angle is known or measured and madeavailable to a calculation process 6 from a source 5 such as a positionsensor. Inflection points and derivatives of the cylinderpressure/angular position curve are calculated at 6. The offset betweenthe expected or theoretical position of a piston according to the knownor measured angular position of the flywheel and the actual local crankangle of the piston is determined at step 7. In a further development ofthe invention a value for the IMEP for the cylinder is calculated atstep 8.

[0040] The calculated value of IMEP for a cylinder is then available foruse as an input to diagnostic and control methods for an engine. TheIMEP of each cylinder of a multi-cylinder engine maybe calculated inthis way. This result may be used in development of new engines,improvement to existing engine and/or monitoring, control and regulationof engines in operation. Mechanical behaviour such as twist of acrankshaft under load may be determined. Operational performance such asthe IMEP value may be used in engine diagnostic and control methods tocompare:

[0041] work output in one cylinder with work output in other cylinders;

[0042] work output according to different engine management settings forone or more cylinders such as valve timing, ignition timing, fuelinjection timing;

[0043] work output for different fuels.

[0044]FIG. 4 shows a schematic diagram for an exemplary arrangement ofapparatus arranged in a system according to a further embodiment of theinvention. The diagram shows a motor 10, a gas pressure sensor 14, acalculating unit 15, a comparation unit 16, and a motor diagnostics andcontrol unit 11. A selection of some possible control actions for themotor are illustrated and shown as motor control actions 12.

[0045] The calculating unit 15 shown in FIG. 4 may be in one or moreparts for calculation firstly of the offset value, and secondly forcalculation of a value for work output, in particular IMEP. The IMEPvalue calculated is compared in comparation unit 16 which may be aseparate component or not. Comparation unit 16 sends typically a signal9 to a motor control unit 11 which may as a result issue one or moremotor control actions 12 such as, for example, to change valve timing,fuel injection timing or ignition timing for one or more cylinders. Thesignal 9 may also be stored for analysis, data-logging or otherpurposes.

[0046] The calculated value for IMEP, based on operational measurementsof cylinder gas pressure per cylinder or cylinders, is then compared toa predetermined value stored in a memory storage unit at 7. Such amemory storage device may be any commercially available device based onsuch as a ROM (Read Only Memory), a Programmable Read Only Memory(PROM), an Eraseable Programmable Read Only Memory (EPROM), in a flashmemory, or in any other non-volatile or permanent storage. Thecomparison at 7 results in a value for IMEP being determined at 8 toeither be within limits, Y, or outside of limits N. A compared IMEP thatis out of limits with the stored value results in an error signal beingsent at 9 to a control unit for the motor. This error signal may then beused in an action to regulate or otherwise adjust the motor.

[0047] As well as, or instead of, relatively fixed Y in-limits or Nout-of-limits, the comparison results may be evaluated to show trends orchanges in IMEP. Such changes or trends may be used to detect forexample an increase in vibration, or of wear, or presence ofcarbonisation or other build-up that have a significance for engineperformance, economy or service life.

[0048] A control action may be taken per cylinder or for multiples ofcylinders. The control action is, typically and not exclusively, one ormore of

[0049] adjusting engine or cylinder ignition timing,

[0050] adjusting fuel injection timing,

[0051] adjusting fuel injection duration,

[0052] adjusting valve timing.

[0053] For example in the case of electrical ignition engines burningfuels such as gasoline control actions may consist of adjusting fuelinjection timing, fuel injection rate, fuel quantity, valve timingand/or compression ratio for engines with variable valve timing and/orcompression ration, or a combination of actions. For example in the caseof diesel engines control actions may consist of adjusting fuelinjection timing, fuel quantity, or a combination of actions.

[0054] Calculation units 5, 6 and comparation unit 7 may be integratedin one or more electronic circuits. However one of more of thesefunctions may equally be included as electronic circuits on boards or inchips according to the known techniques for making miniature controlcircuits. Furthermore, any function of the present invention such as forexample the calculation units 5, 6 or comparator 7 may equally beimplemented as software functions by computer program code, computersoftware or by one or more computer program elements, within a computerprogram in a computer, microprocessor or micro-chip connected to a motoror a control unit for a motor.

[0055] The signal generated by the comparator may take the form of acomputer data signal embodied in a data communication. This data signalthus comprises information about the offset of crank angle of at leastone cylinder of an internal combustion engine. The signal may alsocomprise a corresponding IMEP value. The signal may be communicated to acircuit in the same physical unit, circuit board or chip. Thecommunication may as well by sent by comparator 7 or equivalent over acommunication means to a separate control unit such as diagnostic andcontrol unit 11. The communication means may comprise any communicationdevice or communication network such as a simple wire connection, acable network, a fieldbus, and a mixed network including a wirelesslink. In a generating plant or a ship for example the signal may be sentover a fieldbus. In a car the signal may be sent over a wire or awireless connection. The data signal comprises information about a crankangle offset and/or IMEP, such that upon receipt of said signal acontrol action for the engine may be carried out in respect of saidparameter. Examples of possible control actions such as fuel timing,valve timing etc. have been described above.

[0056] In another embodiment of the invention the method is carried outby computer program code portions contained in a computer software. Acomputer or microprocessor other means hereafter called a processor maybe used to carry out steps of the method by means of one or morecomputer program code portions. The computer program code portionscontains one or more formulae or algorithms according to the method ofthe present invention so that the offset between expected and actualposition in a cylinder can be determined, an accurate value for IMEP etccalculated, and control actions taken dependent on the calculated valuefor IMEP. The computer code portions may be stored in or on any type ofcomputer readable media, including as firmware in devices such as achip, in a ROM (Read Only Memory), a Programmable Read Only Memory(PROM), an Eraseable Programmable Read Only Memory (EPROM), in a flashmemory, or in any other non-volatile or permanent storage. In a best useof the invention, and particularly for larger motors such as for shipsand generators, a pressure sensor of the Cylmate (Trademark) typesupplied by ABB is used. This type of pressure sensor is known to beaccurate, have a long service life and to be affected to a very minimalextent by thermodynamic changes under engine operation.

[0057] In another and favourable embodiment the pressure signal isdigitally sampled at known sample times. The derivative of the pressuresignal is then computed with respect to time, which, if the rotationalspeed is constant, is proportional to the derivative with respect to theflywheel angle. The timing of the inflection points is computed for allcylinders where the pressure is measured in this way, and the timing isthen used together with the known geometry of the crank shaft toestimate the flywheel position at the sample times. The estimation maybe carried out by using an interpolation, such as linear interpolation,or by other known methods. This estimated flywheel position is then usedto compute local crank angles for the pistons at the sample times, andused together with the sampled pressure signals to compute approximatevalues of variables for use as engine diagnostics, such as IMEP. By thismeans the embodiment provides a value for an offset without the need fora measurement of crankshaft angle such as by means of the flywheelangle.

[0058] It is also noted that while the above describes exemplifyingembodiments of the invention, there are several variations andmodifications which may be made to the disclosed solution withoutdeparting from the scope of the present invention as defined in theappended claims.

1. A method to determine a top dead center position for a piston in acylinder of an internal combustion engine comprising one or morecylinders and a crankshaft, where a gas pressure in a cylinder ismeasured together with an angular position of the crankshaft, the methodcomprising: computing an angular position of an inflection point of acompression pressure curve measured as a function of the crankshaftangle, and computing an offset between the angular position and atheoretical value for the angular position.
 2. The method according toclaim 1, wherein the crankshaft angle is obtained from an independentmeasurement of the flywheel angle.
 3. The method according to claim 1,wherein the crankshaft angle is obtained from an independent measurementof the flywheel angle taking twist of the crankshaft into account. 4.The method according to claim 3, wherein the twist is calculated usingthe measured pressure in at least one cylinder, the reciprocatingmasses, and the stiffness of the crank shaft.
 5. The method according toclaim 4, wherein a pressure is estimated for one or more cylinders ofthe engine in which pressure is not measured, which said estimatedpressure is derived from pressure measured in at least one othercylinder, for use in the calculation of the twist.
 6. The methodaccording to claim 3, wherein the twist of the crankshaft is estimatedtaking into account the crankshaft angle measured at both ends of thecrankshaft.
 7. The method according to, claim 4, wherein said stiffnessof the crankshaft is estimated based on said offsets obtained with theengine running at different loads.
 8. The method according to claim 1,wherein said theoretical value for the angular position of theinflection point is obtained according to a formula such as:$\quad \begin{matrix}{a = -} \\\sqrt{\frac{4}{\left( {1 + \lambda} \right)\left( {1 + {2\left( {\gamma - 0.0217} \right)}} \right)\left( {r_{c} - 1} \right)} + \frac{\frac{4}{3}\left( {{- 1} - {3\lambda} + {3\lambda^{2}}} \right)\left( {5 + {4\left( {\gamma - 0.0217} \right)}} \right)}{\left( {1 + \lambda} \right)^{2}\left( {1 + {2\left( {\gamma - 0.0217} \right)}} \right)^{3}\left( {r_{c} - 1} \right)^{2}}}\end{matrix}$

where λ=Relation between crank radius and piston connecting rod length;γ=Quotient between heat capacity at constant pressure and heat capacityat constant volume (polytropic coefficient) for the compressed gas atthe estimated temperature of the gas at the inflection point;r_(c)=Compression ratio of the motor; and α=theoretical inflexion pointin radians.
 9. The method according to claim 8, whrein the temperaturedependence of said polytropic coefficient is taken into account byaddition of a correction term.
 10. The method according to claim 9,wherein the correction term is approximately equal to −0.0217.
 11. Themethod according to claim 8, wherein the theoretical value for theangular position of the inflexion point is obtained by numericallysolving a differential equation for the pressure as a function of thecrank angle.
 12. The method according to claim 1, wherein the furtherstep of calculating a value for diagnostic variables for a cylinder,using said pressure, said crankshaft angle and said offset.
 13. Themethod according to said claim 1, wherein the diagnostic variablesinclude any of the following: Indicated Mean Effective Pressure, heatrelease, heat release rate, angle of maximum derivative, and angle ofmaximum pressure.
 14. The method according to claim 12, furthercomprising comparing said diagnostic variables with a stored value. 15.The method according to claim 13, further comprising issuing a signal toa control unit of the motor dependent on any of said diagnosticvariables.
 16. A method to determine an approximate value of angularposition as a function of time for a piston in a cylinder of an internalcombustion engine comprising one or more cylinders and a crank shaft,where the gas pressure in at least one cylinder is measured as afunction of the time, the method comprising: computing the timing of theinflexion point of each measured compression pressure curve as afunction of time, computing the theoretical value of the angularposition of said piston at said inflection points, and estimating thecrank angle of said piston as a function of time, based on saidinflection point timings and said theoretical value.
 17. The methodaccording to claim 16, wherein the estimate is based on interpolationbetween two of said timings.
 18. The method according to claim 16,wherein the estimate is based on a least squares fit of a polynomial tomore than two of said timings.
 19. A system for determining TDC ofpiston in a cylinder of a motor, said system comprising a gas pressuresensor, and a calculation unit, a memory storage means, wherein thecalculation unit is arranged to calculate an inflection point of apressure curve for gas pressure in the cylinder dependent on crankangle.
 20. The system according to claim 19, wherein the calculationunit comprises a formula to calculate a theoretical value for positionof the piston according to a formula such as: $\quad \begin{matrix}{a = -} \\\sqrt{\frac{4}{\left( {1 + \lambda} \right)\left( {1 + {2\left( {\gamma - 0.0217} \right)}} \right)\left( {r_{c} - 1} \right)} + \frac{\frac{4}{3}\left( {{- 1} - {3\lambda} + {3\lambda^{2}}} \right)\left( {5 + {4\left( {\gamma - 0.0217} \right)}} \right)}{\left( {1 + \lambda} \right)^{2}\left( {1 + {2\left( {\gamma - 0.0217} \right)}} \right)^{3}\left( {r_{c} - 1} \right)^{2}}}\end{matrix}$

where λ=Relation between crank radius and piston connecting rod length;γ=Quotient between heat capacity at constant pressure and heat capacityat constant volume (polytropic coefficient) for the compressed gas atthe estimated temperature of the gas at the inflection point:r_(c)=Compression ratio of the motor; and α=theoretical inflexion pointin radians.
 21. The system according to claim 19, wherein a unit of saidsystem comprises means to generate a signal comprising a value for theoffset.
 22. The system according to claim 19, wherein a unit of saidsystem comprises means to generate a signal comprising a value for workoutput such as IMEP.
 23. A computer program product comprising computerprogram code means which, when run on a computer or processor, willcause the computer or processor to carry out steps of a method accordingto claim
 1. 24. A computer readable medium on which is stored a computerprogram product comprising computer code means to instruct a computer orprocessor carry out the steps of a method according to claim
 1. 25. Useof a system according to claim 19, to calculate an angular position of apiston in a cylinder of an internal combustion engine.
 26. Use of asystem according to claim 19, to calculate a value for IMEP for a pistonin a cylinder of an internal combustion engine.
 27. Use of a computerprogram product according to claim 23 to calculate an angular positionof a piston in a cylinder of an internal combustion engine.
 28. Use of acomputer program product according to claim 23 to calculate a value forIMEP for a cylinder of an internal combustion engine.
 29. Use of acomputer program product according to claim 23 to generate a signal fora control action for control of an internal combustion engine.
 30. Acomputer data signal embodied in a data communication comprisinginformation about a function of gas pressure for a cylinder of aninternal combustion engine, wherein said signal is generated by a systemand provided over a communication means and said information comprisesinformation about an offset of a piston position, and/or a value of workoutput such as IMEP, such that upon receipt of said signal a controlaction for the engine may be carried out.